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Math Help - Orbit of g

  1. #1
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    Orbit of g

    Hi,

    I know that if G acts on set X then the orbit is
    Orbit of g-eq.latex.gif

    But say G=X.

    Does it mean that I put two g's into the above?
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  2. #2
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    Quote Originally Posted by Roland25 View Post
    Hi,

    I know that if G acts on set X then the orbit of x is \color{red}Gx=\{g*x : \ g \in G \}.

    But say G=X. Does it mean that I put two g's into the above?
    if G=X, and x \in X, then \text{orbit}_G(x)=Gx=\{g*x: \ g \in G \}. clearly if * is the same as multiplication operation of G, then Gx=G, for all x \in X=G.
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  3. #3
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    Sorry, I meant G(x) in the above.
    What I mean is that if Gx={g*x : g in G}
    what is Gg if g is in G.
    Would it just be Gg={G} ?
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  4. #4
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    Quote Originally Posted by Roland25 View Post
    Sorry, I meant G(x) in the above.
    What I mean is that if Gx={g*x : g in G}
    what is Gg if g is in G.
    Would it just be Gg={G} ?
    it depends how you define the group action *. if you define that for all x,g \in G: \ g*x=gx, then Gx=G. but if you define g*x=gxg^{-1}, for all g,x \in G, then Gx=\{gxg^{-1}: \ g \in G \},

    which is the conjugacy class of x.
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  5. #5
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    thanks for the help so far.

    I think * was multiplication in my case and so Gx=G kinda seems more appropriate.

    What I still don't get however is what the actual different between Gx and Gg would be if X=G?

    Say X doesn't equal G, would Gx and Gg differ then?

    This is where g is in G
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  6. #6
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    Quote Originally Posted by Roland25 View Post
    thanks for the help so far.

    I think * was multiplication in my case and so Gx=G kinda seems more appropriate.

    What I still don't get however is what the actual different between Gx and Gg would be if X=G?

    Say X doesn't equal G, would Gx and Gg differ then?

    This is where g is in G
    x \in X is fixed and g is any element of G. this should be clear from the definition: Gx= \{g*x: \ g \in G \}.
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